# Caspar Wessel

Caspar Wessel
Born June 8, 1745
Vestby
Died March 25, 1818 (aged 72)
Copenhagen
Nationality  Danish
Norwegian
Alma mater University of Copenhagen
Known for Complex numbers
Complex plane
Vectors
Awards Knight of the Order of the Dannebrog
Scientific career
Fields Mathematics

Caspar Wessel (June 8, 1745, Vestby – March 25, 1818, Copenhagen) was a DanishNorwegian mathematician and cartographer. In 1799, Wessel was the first person to describe the geometrical interpretation of complex numbers as points in the complex plane. He was the younger brother of poet and playwright Johan Herman Wessel.

## Biography

Wessel was born in Jonsrud, Vestby, Akershus, Norway and was one of thirteen children in a family. In 1763, having completed secondary school at Oslo Cathedral School, he went to Denmark for further studies. He attended the University of Copenhagen to study law, but due to financial pressures, could only do so for a year. To survive, he became an assistant land surveyor to his brother and they worked on the Royal Danish Academy of Sciences and Letters' topographical survey of Denmark. This was not enough, however, and he took on extra work as a cartographer. He worked as a surveyor for the rest of his life, stopping only for a sabbatical year in 1778 to finish his law degree.[1] By 1798 had risen to the supervisory role of Royal Inspector of Surveying.[2]

It was the mathematical aspect of surveying that led him to exploring the geometrical significance of complex numbers. His fundamental paper, Om directionens analytiske betegning, was presented in 1797 to the Royal Danish Academy of Sciences and Letters.[3] Since it was in Danish and published in a journal rarely read outside of Denmark, it went unnoticed for nearly a century.[2] The same results were independently rediscovered by Argand in 1806 and Gauss in 1831.[1]

One of the more prominent ideas presented in "On the Analytical Representation of Direction" was that of vectors. Even though this was not Wessel's main intention with the publication, he felt that a geometrical concept of numbers, with length and direction, was needed. Wessel's approach on addition was: "Two straight lines are added if we unite them in such a way that the second line begins where the first one ends and then pass a straight line from the first to the last point of the united lines. This line is the sum of the united lines". This is the same idea as used today when summing vectors.

Wessel's priority to the idea of a complex number as a point in the complex plane is today universally recognised. His paper was re-issued in French translation in 1897,[4] and in English in 1999 as On the analytic representation of direction (eds. Bodil & Lützen).[5]

In 1815, Wessel was made a knight of the Order of the Dannebrog for his contributions to surveying.

## References

1. ^ a b
2. ^ a b Nahin, Paul J. (1998). An Imaginary Tale: The Story of ${\displaystyle {\sqrt {-1}}}$. Princeton University Press. pp. 48–49. ISBN 978-0-691-14600-3.
3. ^ Wessel, Caspar (1799). "Om Directionens analytiske Betegning, et Forsog, anvendt fornemmelig til plane og sphæriske Polygoners Oplosning" [On the analytic representation of direction, an effort applied in particular to the determination of plane and spherical polygons]. Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter (in Danish). Copenhagen: Royal Danish Academy of Sciences and Letters. 5: 469–518.
4. ^ Wessel, Caspar (1799). Essai sur la représentation analytique de la direction [Essay on the Analytic Representation of Direction] (in French). Translated by Zeuthen, H. G. Copenhagen: Royal Danish Academy of Sciences and Letters (published 1897). BNF 31640182t.
5. ^ Wessel, Caspar (1797). Branner, Bodil; Lützen, Jesper, eds. On the analytical representation of direction: an attempt applied chiefly to solving plane and spherical polygons, 1797. Translated by Damhus, Flemming. Copenhagen: C.A. Reitzels (published 1997). ISBN 8778761581. OCLC 43346556.